David Moyes’s side trailed 2-1 at half-time and looked on course for a third successive home game without victory in the league, only for goals from Wayne Rooney and Javier Hernandez to secure a much-needed win.
United remain eight points behind Arsenal, who won 2-0 at Crystal Palace to consolidate their position at the head of the table, but they have closed to within three points of the top four.
Stoke took a fortuitous lead in the fourth minute at Old Trafford when Jonny Evans’s attempted clearance from an Erik Pieters cross cannoned against the prone Peter Crouch and flew into the net.
Robin van Persie equalised in the 43rd minute, tucking home after Asmir Begovic parried a Rooney header, only for Marko Arnautovic to restore the visitors’ lead on the stroke of half-time with a superb free-kick.
Stoke lost the influential Arnautovic to injury in the second half and United capitalised by turning the game around with a pair of goals in the space of three minutes.
Rooney cleverly flicked a looping header over Begovic to equalise in the 78th minute, before substitute Javier Hernandez completed the comeback with a back-post header from Patrice Evra’s teasing cross.
“It was a big challenge to make sure we won the game, but we did it. We showed great energy after the midweek game against Real Sociedad to go until the end,” said United manager Moyes.
“I was worried it wouldn’t come, especially when we were 2-1 down. But I think we deserved it in the end. We kept at it and deserved to score.
“It may be a turning point — we know we have got to do it in style here at Old Trafford.”
With Chelsea not in action until Sunday, when they host Manchester City, Liverpool took advantage to climb above them into second place thanks to a glorious hat-trick by Luis Suarez in a 4-1 win over West Bromwich Albion.
The Uruguayan broke the deadlock with a fine goal in the 12th minute, picking the ball up in a deep position, drifting past one defender and nutmegging another before drilling a low shot past Boaz Myhill.
He added a second five minutes later with an improvised header from just inside the box and completed his first-ever Anfield hat-trick 10 minutes into the second half by nodding home a Steven Gerrard free kick.
James Morrison replied with a penalty in the 66th minute, but Daniel Sturridge restored Liverpool’s three-goal advantage in style 13 minutes from time with an exquisite chip from the left-hand edge of the area.
Liverpool manager Brendan Rodgers praised his two goalscorers, saying: “They’re very hard to contain. Both of them have different qualities and both are goalscorers.
“I thought our midfield three today were excellent. We’re just looking to keep improving, not getting carried away, and the boys at the top end did their job and got the goals.”
Earlier, Arsenal survived the dismissal of goalscorer Mikel Arteta to overcome Palace at Selhurst Park and preserve their status at the top of the standings on the 150th anniversary of the Football Association.
Arteta put the visitors ahead with a 47th-minute penalty, but he was contentiously shown a straight red card with 25 minutes remaining for a last-man foul on former Arsenal man Marouane Chamakh.
Wojciech Szczesny produced two stunning saves to thwart Palace, who parted company with manager Ian Holloway on Wednesday, before Arsenal made sure of victory late on when Olivier Giroud headed home from Aaron Ramsey’s dinked cross.
“I honestly believe (Arteta’s dismissal) was the wrong decision. Why? Because it has to be a clear goalscoring opportunity. It was very far from goal, he was not the last man, and even the foul was, to me, accidental.”
Second-half goals from Romelu Lukaku and Leon Osman earned Everton a 2-0 win at Aston Villa, for whom Lukaku’s Belgian international colleague Christian Benteke saw an early penalty pushed over the bar by Tim Howard.
Norwich City dominated their home game with Cardiff City but had to settle for a 0-0 draw that kept Chris Hughton’s side in the relegation zone.
Southampton, who began the weekend in sixth place, tackle Fulham at St Mary’s in the late game.